Visualizing complex multiplication

Here's an attempt to create a way to visualize elliptic curves with complex multiplication. With any luck, it will also give some insight into why the term "complex multiplication" is used.

An endomorphism of an elliptic curve is a homomorphism that maps points on the curve to other points on the curve. The multiplication-by-n maps

φn: P→ nP

all do this. For most elliptic curves, those are all the endomorphisms that exist, but some elliptic curves have other endomorphisms. Curves like those are said to have complex multiplication, and if we look at a few examples, it's not hard to see why it's called this.

Example 1

For the elliptic curve

y2 = x3 + x

the mapping given by

φ(x,y) = (-x,iy)

is an endomorphism that’s not a multiplication-by-n map. It also has the property that

φ2(P) = φ∘φ(P) = (x,-y) = –P

Because φ2 acts roughly like multiplication by -1, we can think of φ as acting roughly like multiplication by the complex number i.

If we think of this elliptic curve as being parametrized by the Weierstrass ℘-function with periods ω1 and ω2, we find that

ω1 = 1.85407 – 1.85407 i

and

ω2 = 1.85407 + 1.85407 i

Note that ω2 = iω2.

Here’s what the lattice L of points nω1 + mω2 looks like:

Here's what multiplication by i does to a typical point in this lattice:

Note that this lattice has the property that iL = L, so the same multiplication by i that maps points on the elliptic curve to other points on the curve also maps points in the lattice to other points in the lattice.

is an endomorphism that’s not a multiplication-by-n map. It also has the property that

φ3(x,y) = φ∘φ∘φ(x,y) = (x,y)

Because φ3 acts roughly like multiplication by 1, we can think of φ as acting roughly like multiplication by the complex number ξ.

If we think of this elliptic curve as being parametrized by the Weierstrass ℘-function with periods ω1 and ω2, we find that

ω1 = 2.10327 -1.21433 i

and

ω2 = 2.42865 i

Note that we have that ω2 = ξω1.

Here’s what the lattice L of points nω1 + mω2 looks like:

Here's what multiplication by ξ does to a typical point in this lattice:

Note that this lattice has the property that ξL = L, so the same multiplication by ξ that maps points on the elliptic curve to other points on the curve also maps points in the lattice to other points in the lattice.

Example 3

Consider the elliptic curve

y2 = x3 – 4320 x + 96768

If we think of this elliptic curve as being parametrized by the Weierstrass ℘-function with periods ω1 and ω2, we find that

ω1 = -0.296858 i

and

ω2 = 0.419821

Note that ω2 = √2iω1.

Here’s what the lattice L of points nω1 + mω2 looks like:

In this lattice, if we multiply a point z by (√2i)2 we get the point -2z. This looks something like this:

This will also be reflected in the points on the elliptic curve, so this curve has complex multiplication by √2i. In this case, however, it's not at all obvious what the endomorphism is that's not a multiplication-by-n map.